Research in algebra has a wide spectrum in Hungary, researchers ofinternational renown work in various areas of group theory, ring theory,invariant theory, universal algebra, etc. The main goal of the presentproject is to form a research team across several institutions thatwill be able to join the efforts by researchers from different placesand from different areas of algebra in order to make more collaborativeefforts by combining their various specializations to attack a widerange of problems in algebra.

In the spirit of Hungarian mathematics this project is aimed atsolving interesting problems in algebra, in particular the followingones.

In group theory:-Randomness properties of highly symmetric graphs/networks;-Whether every finite lattice is isomorphic to an interval sublatticein the subgroup lattice of a finite group;-When extensions and amalgamated products of sofic groups stay sofic;-The Hidden Subgroup Problem of quantum computing for nilpotent groups.

In the theory of invariants:-The finiteness of the Helly dimension of connected solvable Lie groups;-The completeness of the set of invariants in characteristic 2.

Determination of the matrix type of Cohn algebras.

Criteria guaranteeing that the polynomial ring over a radical ring isalso a radical ring.

Deformations and contractions of Lie algebras and other mathematicalstructures.

We have conducted research in various subfields of abstract algebra (in group theory, in the theory of invariants, in ring theory, in the theory of Lie algebras, in the theory of loops, and in universal algebra). Members of the research team published 58 papers and a book. Many of our publications appeared in leading mathematical periodicals (Annals of Mathematics, Journal of the European Mathematical Society, Journal of Algebra, Journal of Group Theory, Journal of Algebraic Combinatorics, Proceedings of the American Mathematical Society, etc.). We just mention here our three most important results: (1) on the growth in simple groups of Lie type (L. Pyber and E. Szabó); (2) decomposing the discriminant of a 3x3 real symmetric matrix into the sum of 5 squares (M. Domokos); (3) a result concerning a generalization of Higman's conjecture on the number of conjugacy classes in the group of upper unitriangular matrices (Z. Halasi and P. P. Pálfy). Result (1) has been obtained simultaneously by Breuillard, Green, and Tao; it has important implications for expander graphs. Result (2) improves upon a result of Kummer from the middle of nineteenth century; up till now only a decomposition into 7 squares has been known. Result (3) refutes a generalization of Higman's conjecture, hence making the validity of the original conjecture doubtful. The support of OTKA made it possible to employ three talented young researchers as well.